Kappa distributions are widely used in space plasma physics to model velocity distribution functions with heavy tails. Parameter estimation in these distributions is, however, complicated by the fact that the kappa distribution does not belong to the exponential family, so it admits no sufficient statistics and direct maximum likelihood requires numerical optimization without analytically closed-form update equations. Working within the Beck-Cohen superstatistics framework, where a gamma-distributed inverse temperature \(β\) generates the kappa distribution upon marginalization, we treat \(β\) as a latent variable. This hierarchical description restores the exponential family structure that the marginal kappa distribution lacks, and yields an analytically tractable implementation of the expectation-maximization (EM) algorithm whose E-step and M-step admit closed-form expressions in terms of sufficient statistics. Applied to synthetic data drawn from the model, the algorithm converges monotonically to a stationary point of the marginal kappa log-likelihood and recovers the generating parameters consistently across the explored range of \(κ\). EM thus offers a tractable and transparent route to inference in superstatistical systems with local temperature fluctuations.

翻译：在空间等离子体物理中，kappa分布被广泛用于建模具有重尾特性的速度分布函数。然而，此类分布的参数估计存在困难，原因在于kappa分布不属于指数族，导致其不存在充分统计量，且直接进行最大似然估计时需依赖数值优化方法，无法获得解析封闭形式的更新方程。在Beck-Cohen超统计框架中，通过边缘化服从伽马分布的逆温度参数β可生成kappa分布。我们将β视为隐变量，这一分层描述恢复了边缘kappa分布所缺失的指数族结构，并使得期望最大化(EM)算法的实现具有解析可处理性——其E步和M步均可通过充分统计量表示为封闭形式。将该算法应用于模型生成的合成数据时，它能单调收敛至边缘kappa对数似然函数的驻点，并在所考察的κ值范围内一致地恢复生成参数。因此，EM算法为具有局部温度波动的超统计系统提供了一条可处理且透明的推断途径。